Generalized Slow Roll and CMB Constraints on the Inflaton Poten8al 1 Cora Dvorkin University of Chicago/KICP June 2010, Great Lakes Cosmology Workshop References: C. Dvorkin and W. Hu, astroph/0910.2237, PRD (2009). M. Mortonson, C. Dvorkin, H.V.Peiris, W.Hu, astroph/0903.4920 , PRD (2009). C. Dvorkin and W. Hu (2010, in preparaHon).
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Generalized Slow Roll and
CMB Constraints on the Inflaton Poten8al
1
Cora Dvorkin University of Chicago/KICP
June 2010, Great Lakes Cosmology Workshop
References: -‐ C. Dvorkin and W. Hu, astro-‐ph/0910.2237, PRD (2009). -‐ M. Mortonson, C. Dvorkin, H.V.Peiris, W.Hu, astro-‐ph/0903.4920 , PRD (2009). -‐ C. Dvorkin and W. Hu (2010, in preparaHon).
Ordinary slow roll (brief background)
Technique for compuMng the iniMal curvature power spectrum for inflaMonary models where the scalar field potenMal is sufficiently flat and slowly varying.
�H ≡ 12
�φ̇
H
�2
ηH ≡ −�
φ̈
Hφ̇
�
δ2 ≡...φ
H2φ̇
Slow-‐roll parameters:
Linked to the shape of the potenHal
<< 1 and slowly varying
Slow roll approximaMon: ∆2R ≈
�(1− (2C + 1)�H − CηH)
H2
2π|φ̇|
�2
kη≈1
Why go further?
• These models require order unity variaMons in the curvature power spectrum: slow-‐roll parameters are not necessarily small or slowly varying.
3
• Glitches in the temperature power spectrum of the CMB have led to interest in exploring models with features in the inflaton potenMal.
L. Covi, J.Hamann, A. Melchiorri, A. Slozar and I. Sorbera, (2006). M. Mortonson, C. Dvorkin, H.V. Peiris and W. Hu, PRD (2009).
InflaMonary features A step in the inflaMonary potenMal generates an oscillatory iniMal
curvature power spectrum.
The slow-‐roll parameters are neither small nor slowly varying. 4
C. Dvorkin, W.Hu, PRD (2009)
INFLATIONARY POTENTIAL POWER SPECTRUM
TEMPERATURE
POLARIZATION
INFLATIONARY POTENTIAL POWER SPECTRUM
TEMPERATURE
POLARIZATION
OBSERVABLES
How does the Generalized Slow Roll approximaMon work?
7
• Field equaMon:
• Perfect slow roll:
• GSR approximaMon:
d2y
dx2+
�1− 2
x2
�y =
g(lnx)x2
y
d2y0
dx2+
�1− 2
x2
�y0 = 0
d2y
dx2+
�1− 2
x2
�y =
g(lnx)x2
y0
SoluMon can be constructed with Green funcMon technique.
Source funcMon (linear in slow-‐roll parameters)
(y =√
2kuk; x = kη)
E. Stewart, PRD (2002).
How does the Generalized Slow Roll approximaMon work?
8
• Field equaMon:
• Perfect slow roll:
• GSR approximaMon:
d2y
dx2+
�1− 2
x2
�y =
g(lnx)x2
y
d2y0
dx2+
�1− 2
x2
�y0 = 0
d2y
dx2+
�1− 2
x2
�y =
g(lnx)x2
y0
SoluMon can be constructed with Green funcMon technique.
• Nodes in the power spectrum. • Curvature is not constant for modes outside the horizon.
BUT…
Source funcMon (linear in slow-‐roll parameters)
(y =√
2kuk; x = kη)
E. Stewart, PRD (2002)
Generalized Slow Roll for Large DeviaMons
9
The curvature power spectrum only depends on a single source funcMon:
C. Dvorkin, W. Hu, PRD (2009)
ln ∆2R(k) = G(ln ηmin) +
� ηmax
ηmin
dη
ηW (kη)G�(ln η)
+ ln
�1 +
12
�� ηmax
ηmin
dη
ηX(kη)G�(ln η)
�2�
Generalized Slow Roll for Large DeviaMons
10
The curvature power spectrum only depends on a single source funcMon:
C. Dvorkin, W. Hu, PRD (2009)
ln ∆2R(k) = G(ln ηmin) +
� ηmax
ηmin
dη
ηW (kη)G�(ln η)
+ ln
�1 +
12
�� ηmax
ηmin
dη
ηX(kη)G�(ln η)
�2�
Constant curvature for modes outside the horizon. Well controlled at large values of the source: percent level errors. (Mme integrals are small even if the source is not small everywhere).
Simple to relate to the Inflaton PotenMal: G� ≈ 3�
V,φ
V
�2
− 2�
V,φφ
V
�
Second order Generalized Slow Roll: Well controlled
11
TEMPERATURE POLARIZATION
M.Mortonson, C. Dvorkin, H.V.Peiris, W.Hu, PRD (2009)
PolarizaHon can test the feature hypothesis at 2.5 sigma with Planck, 5-‐6 sigma with CMBpol.
Second order Generalized Slow Roll: Well controlled
12
TEMPERATURE POLARIZATION
C. Dvorkin, W. Hu, PRD (2009)
13
We can use features in the power spectrum to directly constrain features in the potenHal.
Power spectrum
Source
PotenMal
Model independent constraints (PC’s) (Work in progress)
Principal components: Basis for a complete representaMon of observable properMes of the source funcMon.
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GSR approximaHon allows us to go beyond specific infla8onary models and think of the data as directly constraining the source funcHon.
C. Dvorkin, W. Hu (2010, in prepara8on)
G� = 1− ns +N�
a=1
maSa(ln η)
WMAP7 constraints from MCMC’s (Work in progress)
15
C. Dvorkin, W. Hu (2010, in prepara8on)
As a by-‐product: We parallelized the
WMAP likelihood evaluaMon for mulM-‐core systems
and we improved its speed by ~ 5*Ncore
Summary
• The Generalized Slow Roll approximaMon is accurate at the percent level for order unity deviaMons in the CMB temperature and polarizaMon power spectra.
• There is a single source funcMon responsible for observed features and it is simply related to the local slope and curvature of the Inflaton PotenMal.
• ApplicaMons (work in progress): Use this technique for model independent constraints on the
Inflaton PotenMal: think of the data as directly constraining the source funcMon.